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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{多元统计分析练习2}
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\date{2024 年 3 月 12 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
%例子2.1.1
%一副麻将牌除``花''之外有34种不同的牌，每种牌有相同的4张，共有 $34\times 4=136$ 张。
一副扑克牌除``王''之外有13种不同的牌，每种牌有 $\diamondsuit \, \heartsuit \, \clubsuit \, \spadesuit$ 四张，共有 $13\times 4=52$ 张。 
打牌开始时，每人摸牌13张，摸到13种不同的牌的概率是多少？

\vspace{0.2cm}

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\item  %Problem 02
%例子2.1.2
设随机向量 $x=(x_1,x_2)'$ 有概率密度 
\begin{eqnarray*}
f(x_1,x_2) = \left\{ \begin{array}{ll}
\frac{6}{5}x_1^2(4x_1x_2+1), & 0< x_1< 1, 0< x_2< 1, \\ 
0, & \mathrm{其它}. 
\end{array}\right. 
\end{eqnarray*}
试求条件密度 $f(x_1\mid x_2)$ 和 $f(x_2\mid x_1)$. 

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\item  %Problem 03
设随机向量 $x=(x_1,x_2,x_3)'$ 的数学期望和协方差矩阵分别是
\begin{eqnarray*}
\mu=\begin{pmatrix}5 \\ -2 \\ 7 \end{pmatrix}, \,\,\, 
\mu=\begin{pmatrix} 4&1&2 \\ 1&9&-3 \\ 2&-3&25 \end{pmatrix}.
\end{eqnarray*}
\begin{enumerate}[label={(\arabic*)}]
\item  求 $x$ 的相关矩阵 $R$. 
\item  设 $y_1=2x_1-x_2+4x_3, y_2=x_2-x_3, y_3=x_1+3x_2-2x_3$, 求 $y=(y_1,y_2,y_3)'$ 的数学期望和协方差矩阵。
\end{enumerate}

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\item  %Problem 04
设随机向量 $x_1,x_2$ 都是 $p$ 维的，设 $x=\begin{pmatrix}x_1 \\ x_2 \end{pmatrix}$ 的协方差矩阵为 $\Sigma= \begin{pmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{pmatrix}$. 求 $x_1+x_2$ 与 $x_1-x_2$ 的协方差矩阵。 

\vspace{0.2cm}

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\item  %Problem 05
设 $x,y$ 是总体 $\pi$ 中的两个样品，该总体的均值和协方差矩阵分别为 $\mu$ 和 $\Sigma$, 定义 $x$ 和 $y$ 之间的平方马氏距离为 $$d^2(x,y) = (x-y)'\Sigma^{-1}(x-y). $$

设总体是 $p=2$ 维随机向量，设 $c$ 是一个正数，证明 到 $\mu$ 马氏距离为固定值 $c$ 的样品 $x$ 的集合 
$$\{x: (x-\mu)'\Sigma^{-1}(x-\mu)=c^2\}$$
 是一个椭圆。

\vspace{0.2cm}

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\item  %Problem 06
设随机变量 $x$ 服从均匀分布 $U(0,1)$, 设 $\lambda>0$, 求随机变量 $y=-\frac{1}{\lambda} \ln x$ 的分布。

\vspace{0.2cm}

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\item  %Problem 07
定义随机变量 $x$ 的特征函数为 $$\varphi(t) = E(e^{itx}), -\infty<t<\infty. $$
设随机变量 $x$ 服从标准正态分布 $N(0,1)$, 用特征函数求 $y=\mu+\sigma x$ 的分布。

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\item  %Problem 08
设 $p$ 维随机向量 $x$ 的均值向量和协方差矩阵分别为 $\mu$ 和 $\Sigma$. 
证明 $$E(xx')=\Sigma + \mu\mu'. $$ 

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%\item  %Problem 09
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\end{enumerate}


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\end{document}

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